Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^2$

TL;DR

This paper investigates the behavior of geodesics in two-dimensional first-passage percolation on various subgraphs of z^2, establishing conditions for their convergence and properties of the limiting geodesic structure.

Contribution

It proves the almost sure convergence of geodesics to boundary points on a broad class of subgraphs and characterizes the structure of the limiting geodesic graph in the half-plane.

Findings

Geodesics from any point to boundary vertices converge almost surely.

Existence of a limiting Busemann function for all passage-time configurations.

The limiting geodesic graph in the half-plane has one topological end, with all geodesics coalescing.

Abstract

It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this question for first-passage percolation on a wide class of subgraphs of $\mathbb {Z}^2$: those whose vertex set is infinite and connected with an infinite connected complement. This includes, for instance, slit planes, half-planes and sectors. Writing $x_n$ for the sequence of boundary vertices, we show that the sequence of geodesics from any point to $x_n$ has an almost sure limit assuming only existence of finite geodesics. For all passage-time configurations, we show existence of a limiting Busemann function. Specializing to the case of the half-plane, we prove that the limiting geodesic graph has one topological end; that is, all its infinite geodesics coalesce, and there are no backward infinite paths. To do this, we prove in the Appendix existence of geodesics for all product measures in our domains and remove the moment assumption of the Wehr-Woo theorem on absence of bigeodesics in the half-plane.